In this study, a new regression method called Kappa regression is introduced to model conditional probabilities. The regression function is based on Dombi’s Kappa function, which is well known in fuzzy theory. Here, we discuss how the Kappa function relates to the Logistic function as well as how it can be used to approximate the Logistic function. We introduce the so-called Generalized Kappa Differential Equation and show that both the Kappa and the Logistic functions can be derived from it. Kappa regression, like binary Logistic regression, models the conditional probability of the event that a dichotomous random variable takes a particular value at a given value of an explanatory variable. This new regression method may be viewed as an alternative to binary Logistic regression, but while in binary Logistic regression the explanatory variable is defined over the entire Euclidean space, in the Kappa regression model the predictor variable is defined over a bounded subset of the Euclidean space. We will also show that asymptotic Kappa regression is Logistic regression. The advantages of this novel method are demonstrated by means of an example, and afterwards some implications are discussed.

中文翻译：

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Kappa 回归：Logistic 回归的替代方法

在这项研究中，引入了一种称为 Kappa 回归的新回归方法来模拟条件概率。回归函数基于模糊理论中众所周知的 Dombi 的 Kappa 函数。在这里，我们讨论 Kappa 函数如何与 Logistic 函数相关，以及如何使用它来逼近 Logistic 函数。我们引入了所谓的广义 Kappa 微分方程，并表明 Kappa 和 Logistic 函数都可以从中导出。Kappa 回归与二元 Logistic 回归一样，对二分随机变量在解释变量的给定值处取特定值的事件的条件概率进行建模。这种新的回归方法可以被视为二元 Logistic 回归的替代方法，但是在二元逻辑回归中，解释变量是在整个欧几里得空间上定义的，而在 Kappa 回归模型中，预测变量是在欧几里得空间的有界子集上定义的。我们还将证明渐近 Kappa 回归是 Logistic 回归。通过一个例子证明了这种新方法的优点，然后讨论了一些启示。